Step |
Hyp |
Ref |
Expression |
1 |
|
gsumzadd.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
gsumzadd.0 |
⊢ 0 = ( 0g ‘ 𝐺 ) |
3 |
|
gsumzadd.p |
⊢ + = ( +g ‘ 𝐺 ) |
4 |
|
gsumzadd.z |
⊢ 𝑍 = ( Cntz ‘ 𝐺 ) |
5 |
|
gsumzadd.g |
⊢ ( 𝜑 → 𝐺 ∈ Mnd ) |
6 |
|
gsumzadd.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
7 |
|
gsumzadd.fn |
⊢ ( 𝜑 → 𝐹 finSupp 0 ) |
8 |
|
gsumzadd.hn |
⊢ ( 𝜑 → 𝐻 finSupp 0 ) |
9 |
|
gsumzaddlem.w |
⊢ 𝑊 = ( ( 𝐹 ∪ 𝐻 ) supp 0 ) |
10 |
|
gsumzaddlem.f |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
11 |
|
gsumzaddlem.h |
⊢ ( 𝜑 → 𝐻 : 𝐴 ⟶ 𝐵 ) |
12 |
|
gsumzaddlem.1 |
⊢ ( 𝜑 → ran 𝐹 ⊆ ( 𝑍 ‘ ran 𝐹 ) ) |
13 |
|
gsumzaddlem.2 |
⊢ ( 𝜑 → ran 𝐻 ⊆ ( 𝑍 ‘ ran 𝐻 ) ) |
14 |
|
gsumzaddlem.3 |
⊢ ( 𝜑 → ran ( 𝐹 ∘f + 𝐻 ) ⊆ ( 𝑍 ‘ ran ( 𝐹 ∘f + 𝐻 ) ) ) |
15 |
|
gsumzaddlem.4 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝐴 ∧ 𝑘 ∈ ( 𝐴 ∖ 𝑥 ) ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ 𝑥 ) ) } ) ) |
16 |
1 2
|
mndidcl |
⊢ ( 𝐺 ∈ Mnd → 0 ∈ 𝐵 ) |
17 |
5 16
|
syl |
⊢ ( 𝜑 → 0 ∈ 𝐵 ) |
18 |
1 3 2
|
mndlid |
⊢ ( ( 𝐺 ∈ Mnd ∧ 0 ∈ 𝐵 ) → ( 0 + 0 ) = 0 ) |
19 |
5 17 18
|
syl2anc |
⊢ ( 𝜑 → ( 0 + 0 ) = 0 ) |
20 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( 0 + 0 ) = 0 ) |
21 |
2
|
fvexi |
⊢ 0 ∈ V |
22 |
21
|
a1i |
⊢ ( 𝜑 → 0 ∈ V ) |
23 |
11 6
|
fexd |
⊢ ( 𝜑 → 𝐻 ∈ V ) |
24 |
23
|
suppun |
⊢ ( 𝜑 → ( 𝐹 supp 0 ) ⊆ ( ( 𝐹 ∪ 𝐻 ) supp 0 ) ) |
25 |
24 9
|
sseqtrrdi |
⊢ ( 𝜑 → ( 𝐹 supp 0 ) ⊆ 𝑊 ) |
26 |
10 6 22 25
|
gsumcllem |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 0 ) ) |
27 |
26
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ 0 ) ) ) |
28 |
2
|
gsumz |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉 ) → ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ 0 ) ) = 0 ) |
29 |
5 6 28
|
syl2anc |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ 0 ) ) = 0 ) |
30 |
29
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ 0 ) ) = 0 ) |
31 |
27 30
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( 𝐺 Σg 𝐹 ) = 0 ) |
32 |
10 6
|
fexd |
⊢ ( 𝜑 → 𝐹 ∈ V ) |
33 |
32
|
suppun |
⊢ ( 𝜑 → ( 𝐻 supp 0 ) ⊆ ( ( 𝐻 ∪ 𝐹 ) supp 0 ) ) |
34 |
|
uncom |
⊢ ( 𝐹 ∪ 𝐻 ) = ( 𝐻 ∪ 𝐹 ) |
35 |
34
|
oveq1i |
⊢ ( ( 𝐹 ∪ 𝐻 ) supp 0 ) = ( ( 𝐻 ∪ 𝐹 ) supp 0 ) |
36 |
33 35
|
sseqtrrdi |
⊢ ( 𝜑 → ( 𝐻 supp 0 ) ⊆ ( ( 𝐹 ∪ 𝐻 ) supp 0 ) ) |
37 |
36 9
|
sseqtrrdi |
⊢ ( 𝜑 → ( 𝐻 supp 0 ) ⊆ 𝑊 ) |
38 |
11 6 22 37
|
gsumcllem |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → 𝐻 = ( 𝑥 ∈ 𝐴 ↦ 0 ) ) |
39 |
38
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( 𝐺 Σg 𝐻 ) = ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ 0 ) ) ) |
40 |
39 30
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( 𝐺 Σg 𝐻 ) = 0 ) |
41 |
31 40
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) = ( 0 + 0 ) ) |
42 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → 𝐴 ∈ 𝑉 ) |
43 |
17
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑊 = ∅ ) ∧ 𝑥 ∈ 𝐴 ) → 0 ∈ 𝐵 ) |
44 |
42 43 43 26 38
|
offval2 |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( 𝐹 ∘f + 𝐻 ) = ( 𝑥 ∈ 𝐴 ↦ ( 0 + 0 ) ) ) |
45 |
20
|
mpteq2dv |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( 𝑥 ∈ 𝐴 ↦ ( 0 + 0 ) ) = ( 𝑥 ∈ 𝐴 ↦ 0 ) ) |
46 |
44 45
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( 𝐹 ∘f + 𝐻 ) = ( 𝑥 ∈ 𝐴 ↦ 0 ) ) |
47 |
46
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( 𝐺 Σg ( 𝐹 ∘f + 𝐻 ) ) = ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ 0 ) ) ) |
48 |
47 30
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( 𝐺 Σg ( 𝐹 ∘f + 𝐻 ) ) = 0 ) |
49 |
20 41 48
|
3eqtr4rd |
⊢ ( ( 𝜑 ∧ 𝑊 = ∅ ) → ( 𝐺 Σg ( 𝐹 ∘f + 𝐻 ) ) = ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) ) |
50 |
49
|
ex |
⊢ ( 𝜑 → ( 𝑊 = ∅ → ( 𝐺 Σg ( 𝐹 ∘f + 𝐻 ) ) = ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) ) ) |
51 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → 𝐺 ∈ Mnd ) |
52 |
1 3
|
mndcl |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) → ( 𝑧 + 𝑤 ) ∈ 𝐵 ) |
53 |
52
|
3expb |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) → ( 𝑧 + 𝑤 ) ∈ 𝐵 ) |
54 |
51 53
|
sylan |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) → ( 𝑧 + 𝑤 ) ∈ 𝐵 ) |
55 |
54
|
caovclg |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝐵 ) |
56 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( ♯ ‘ 𝑊 ) ∈ ℕ ) |
57 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
58 |
56 57
|
eleqtrdi |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( ♯ ‘ 𝑊 ) ∈ ( ℤ≥ ‘ 1 ) ) |
59 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → 𝐹 : 𝐴 ⟶ 𝐵 ) |
60 |
|
f1of1 |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 → 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1→ 𝑊 ) |
61 |
60
|
ad2antll |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1→ 𝑊 ) |
62 |
|
suppssdm |
⊢ ( ( 𝐹 ∪ 𝐻 ) supp 0 ) ⊆ dom ( 𝐹 ∪ 𝐻 ) |
63 |
62
|
a1i |
⊢ ( 𝜑 → ( ( 𝐹 ∪ 𝐻 ) supp 0 ) ⊆ dom ( 𝐹 ∪ 𝐻 ) ) |
64 |
9
|
a1i |
⊢ ( 𝜑 → 𝑊 = ( ( 𝐹 ∪ 𝐻 ) supp 0 ) ) |
65 |
|
dmun |
⊢ dom ( 𝐹 ∪ 𝐻 ) = ( dom 𝐹 ∪ dom 𝐻 ) |
66 |
10
|
fdmd |
⊢ ( 𝜑 → dom 𝐹 = 𝐴 ) |
67 |
11
|
fdmd |
⊢ ( 𝜑 → dom 𝐻 = 𝐴 ) |
68 |
66 67
|
uneq12d |
⊢ ( 𝜑 → ( dom 𝐹 ∪ dom 𝐻 ) = ( 𝐴 ∪ 𝐴 ) ) |
69 |
|
unidm |
⊢ ( 𝐴 ∪ 𝐴 ) = 𝐴 |
70 |
68 69
|
eqtrdi |
⊢ ( 𝜑 → ( dom 𝐹 ∪ dom 𝐻 ) = 𝐴 ) |
71 |
65 70
|
eqtr2id |
⊢ ( 𝜑 → 𝐴 = dom ( 𝐹 ∪ 𝐻 ) ) |
72 |
63 64 71
|
3sstr4d |
⊢ ( 𝜑 → 𝑊 ⊆ 𝐴 ) |
73 |
72
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → 𝑊 ⊆ 𝐴 ) |
74 |
|
f1ss |
⊢ ( ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1→ 𝑊 ∧ 𝑊 ⊆ 𝐴 ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1→ 𝐴 ) |
75 |
61 73 74
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1→ 𝐴 ) |
76 |
|
f1f |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1→ 𝐴 → 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐴 ) |
77 |
75 76
|
syl |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐴 ) |
78 |
|
fco |
⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐴 ) → ( 𝐹 ∘ 𝑓 ) : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐵 ) |
79 |
59 77 78
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( 𝐹 ∘ 𝑓 ) : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐵 ) |
80 |
79
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐹 ∘ 𝑓 ) ‘ 𝑘 ) ∈ 𝐵 ) |
81 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → 𝐻 : 𝐴 ⟶ 𝐵 ) |
82 |
|
fco |
⊢ ( ( 𝐻 : 𝐴 ⟶ 𝐵 ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐴 ) → ( 𝐻 ∘ 𝑓 ) : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐵 ) |
83 |
81 77 82
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( 𝐻 ∘ 𝑓 ) : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐵 ) |
84 |
83
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐻 ∘ 𝑓 ) ‘ 𝑘 ) ∈ 𝐵 ) |
85 |
59
|
ffnd |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → 𝐹 Fn 𝐴 ) |
86 |
81
|
ffnd |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → 𝐻 Fn 𝐴 ) |
87 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → 𝐴 ∈ 𝑉 ) |
88 |
|
ovexd |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( 1 ... ( ♯ ‘ 𝑊 ) ) ∈ V ) |
89 |
|
inidm |
⊢ ( 𝐴 ∩ 𝐴 ) = 𝐴 |
90 |
85 86 77 87 87 88 89
|
ofco |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( ( 𝐹 ∘f + 𝐻 ) ∘ 𝑓 ) = ( ( 𝐹 ∘ 𝑓 ) ∘f + ( 𝐻 ∘ 𝑓 ) ) ) |
91 |
90
|
fveq1d |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( ( ( 𝐹 ∘f + 𝐻 ) ∘ 𝑓 ) ‘ 𝑘 ) = ( ( ( 𝐹 ∘ 𝑓 ) ∘f + ( 𝐻 ∘ 𝑓 ) ) ‘ 𝑘 ) ) |
92 |
91
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( ( 𝐹 ∘f + 𝐻 ) ∘ 𝑓 ) ‘ 𝑘 ) = ( ( ( 𝐹 ∘ 𝑓 ) ∘f + ( 𝐻 ∘ 𝑓 ) ) ‘ 𝑘 ) ) |
93 |
|
fnfco |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐴 ) → ( 𝐹 ∘ 𝑓 ) Fn ( 1 ... ( ♯ ‘ 𝑊 ) ) ) |
94 |
85 77 93
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( 𝐹 ∘ 𝑓 ) Fn ( 1 ... ( ♯ ‘ 𝑊 ) ) ) |
95 |
|
fnfco |
⊢ ( ( 𝐻 Fn 𝐴 ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐴 ) → ( 𝐻 ∘ 𝑓 ) Fn ( 1 ... ( ♯ ‘ 𝑊 ) ) ) |
96 |
86 77 95
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( 𝐻 ∘ 𝑓 ) Fn ( 1 ... ( ♯ ‘ 𝑊 ) ) ) |
97 |
|
inidm |
⊢ ( ( 1 ... ( ♯ ‘ 𝑊 ) ) ∩ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) = ( 1 ... ( ♯ ‘ 𝑊 ) ) |
98 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐹 ∘ 𝑓 ) ‘ 𝑘 ) = ( ( 𝐹 ∘ 𝑓 ) ‘ 𝑘 ) ) |
99 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐻 ∘ 𝑓 ) ‘ 𝑘 ) = ( ( 𝐻 ∘ 𝑓 ) ‘ 𝑘 ) ) |
100 |
94 96 88 88 97 98 99
|
ofval |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( ( 𝐹 ∘ 𝑓 ) ∘f + ( 𝐻 ∘ 𝑓 ) ) ‘ 𝑘 ) = ( ( ( 𝐹 ∘ 𝑓 ) ‘ 𝑘 ) + ( ( 𝐻 ∘ 𝑓 ) ‘ 𝑘 ) ) ) |
101 |
92 100
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( ( 𝐹 ∘f + 𝐻 ) ∘ 𝑓 ) ‘ 𝑘 ) = ( ( ( 𝐹 ∘ 𝑓 ) ‘ 𝑘 ) + ( ( 𝐻 ∘ 𝑓 ) ‘ 𝑘 ) ) ) |
102 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → 𝐺 ∈ Mnd ) |
103 |
|
elfzouz |
⊢ ( 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) → 𝑛 ∈ ( ℤ≥ ‘ 1 ) ) |
104 |
103
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → 𝑛 ∈ ( ℤ≥ ‘ 1 ) ) |
105 |
|
elfzouz2 |
⊢ ( 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) → ( ♯ ‘ 𝑊 ) ∈ ( ℤ≥ ‘ 𝑛 ) ) |
106 |
105
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ 𝑊 ) ∈ ( ℤ≥ ‘ 𝑛 ) ) |
107 |
|
fzss2 |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ( ℤ≥ ‘ 𝑛 ) → ( 1 ... 𝑛 ) ⊆ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) |
108 |
106 107
|
syl |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 1 ... 𝑛 ) ⊆ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) |
109 |
108
|
sselda |
⊢ ( ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑘 ∈ ( 1 ... 𝑛 ) ) → 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) |
110 |
80
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐹 ∘ 𝑓 ) ‘ 𝑘 ) ∈ 𝐵 ) |
111 |
109 110
|
syldan |
⊢ ( ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑘 ∈ ( 1 ... 𝑛 ) ) → ( ( 𝐹 ∘ 𝑓 ) ‘ 𝑘 ) ∈ 𝐵 ) |
112 |
1 3
|
mndcl |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑘 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑘 + 𝑥 ) ∈ 𝐵 ) |
113 |
112
|
3expb |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑘 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ) → ( 𝑘 + 𝑥 ) ∈ 𝐵 ) |
114 |
102 113
|
sylan |
⊢ ( ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) ∧ ( 𝑘 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ) → ( 𝑘 + 𝑥 ) ∈ 𝐵 ) |
115 |
104 111 114
|
seqcl |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( seq 1 ( + , ( 𝐹 ∘ 𝑓 ) ) ‘ 𝑛 ) ∈ 𝐵 ) |
116 |
84
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐻 ∘ 𝑓 ) ‘ 𝑘 ) ∈ 𝐵 ) |
117 |
109 116
|
syldan |
⊢ ( ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑘 ∈ ( 1 ... 𝑛 ) ) → ( ( 𝐻 ∘ 𝑓 ) ‘ 𝑘 ) ∈ 𝐵 ) |
118 |
104 117 114
|
seqcl |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( seq 1 ( + , ( 𝐻 ∘ 𝑓 ) ) ‘ 𝑛 ) ∈ 𝐵 ) |
119 |
|
fzofzp1 |
⊢ ( 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) → ( 𝑛 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) |
120 |
|
ffvelrn |
⊢ ( ( ( 𝐹 ∘ 𝑓 ) : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐵 ∧ ( 𝑛 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐹 ∘ 𝑓 ) ‘ ( 𝑛 + 1 ) ) ∈ 𝐵 ) |
121 |
79 119 120
|
syl2an |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐹 ∘ 𝑓 ) ‘ ( 𝑛 + 1 ) ) ∈ 𝐵 ) |
122 |
|
ffvelrn |
⊢ ( ( ( 𝐻 ∘ 𝑓 ) : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐵 ∧ ( 𝑛 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐻 ∘ 𝑓 ) ‘ ( 𝑛 + 1 ) ) ∈ 𝐵 ) |
123 |
83 119 122
|
syl2an |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐻 ∘ 𝑓 ) ‘ ( 𝑛 + 1 ) ) ∈ 𝐵 ) |
124 |
|
fvco3 |
⊢ ( ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐴 ∧ ( 𝑛 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐹 ∘ 𝑓 ) ‘ ( 𝑛 + 1 ) ) = ( 𝐹 ‘ ( 𝑓 ‘ ( 𝑛 + 1 ) ) ) ) |
125 |
77 119 124
|
syl2an |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐹 ∘ 𝑓 ) ‘ ( 𝑛 + 1 ) ) = ( 𝐹 ‘ ( 𝑓 ‘ ( 𝑛 + 1 ) ) ) ) |
126 |
|
fveq2 |
⊢ ( 𝑘 = ( 𝑓 ‘ ( 𝑛 + 1 ) ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ ( 𝑓 ‘ ( 𝑛 + 1 ) ) ) ) |
127 |
126
|
eleq1d |
⊢ ( 𝑘 = ( 𝑓 ‘ ( 𝑛 + 1 ) ) → ( ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) ↔ ( 𝐹 ‘ ( 𝑓 ‘ ( 𝑛 + 1 ) ) ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) ) ) |
128 |
15
|
expr |
⊢ ( ( 𝜑 ∧ 𝑥 ⊆ 𝐴 ) → ( 𝑘 ∈ ( 𝐴 ∖ 𝑥 ) → ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ 𝑥 ) ) } ) ) ) |
129 |
128
|
ralrimiv |
⊢ ( ( 𝜑 ∧ 𝑥 ⊆ 𝐴 ) → ∀ 𝑘 ∈ ( 𝐴 ∖ 𝑥 ) ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ 𝑥 ) ) } ) ) |
130 |
129
|
ex |
⊢ ( 𝜑 → ( 𝑥 ⊆ 𝐴 → ∀ 𝑘 ∈ ( 𝐴 ∖ 𝑥 ) ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ 𝑥 ) ) } ) ) ) |
131 |
130
|
alrimiv |
⊢ ( 𝜑 → ∀ 𝑥 ( 𝑥 ⊆ 𝐴 → ∀ 𝑘 ∈ ( 𝐴 ∖ 𝑥 ) ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ 𝑥 ) ) } ) ) ) |
132 |
131
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ∀ 𝑥 ( 𝑥 ⊆ 𝐴 → ∀ 𝑘 ∈ ( 𝐴 ∖ 𝑥 ) ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ 𝑥 ) ) } ) ) ) |
133 |
|
imassrn |
⊢ ( 𝑓 “ ( 1 ... 𝑛 ) ) ⊆ ran 𝑓 |
134 |
77
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐴 ) |
135 |
134
|
frnd |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ran 𝑓 ⊆ 𝐴 ) |
136 |
133 135
|
sstrid |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑓 “ ( 1 ... 𝑛 ) ) ⊆ 𝐴 ) |
137 |
|
vex |
⊢ 𝑓 ∈ V |
138 |
137
|
imaex |
⊢ ( 𝑓 “ ( 1 ... 𝑛 ) ) ∈ V |
139 |
|
sseq1 |
⊢ ( 𝑥 = ( 𝑓 “ ( 1 ... 𝑛 ) ) → ( 𝑥 ⊆ 𝐴 ↔ ( 𝑓 “ ( 1 ... 𝑛 ) ) ⊆ 𝐴 ) ) |
140 |
|
difeq2 |
⊢ ( 𝑥 = ( 𝑓 “ ( 1 ... 𝑛 ) ) → ( 𝐴 ∖ 𝑥 ) = ( 𝐴 ∖ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) |
141 |
|
reseq2 |
⊢ ( 𝑥 = ( 𝑓 “ ( 1 ... 𝑛 ) ) → ( 𝐻 ↾ 𝑥 ) = ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) |
142 |
141
|
oveq2d |
⊢ ( 𝑥 = ( 𝑓 “ ( 1 ... 𝑛 ) ) → ( 𝐺 Σg ( 𝐻 ↾ 𝑥 ) ) = ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) ) |
143 |
142
|
sneqd |
⊢ ( 𝑥 = ( 𝑓 “ ( 1 ... 𝑛 ) ) → { ( 𝐺 Σg ( 𝐻 ↾ 𝑥 ) ) } = { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) |
144 |
143
|
fveq2d |
⊢ ( 𝑥 = ( 𝑓 “ ( 1 ... 𝑛 ) ) → ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ 𝑥 ) ) } ) = ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) ) |
145 |
144
|
eleq2d |
⊢ ( 𝑥 = ( 𝑓 “ ( 1 ... 𝑛 ) ) → ( ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ 𝑥 ) ) } ) ↔ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) ) ) |
146 |
140 145
|
raleqbidv |
⊢ ( 𝑥 = ( 𝑓 “ ( 1 ... 𝑛 ) ) → ( ∀ 𝑘 ∈ ( 𝐴 ∖ 𝑥 ) ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ 𝑥 ) ) } ) ↔ ∀ 𝑘 ∈ ( 𝐴 ∖ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) ) ) |
147 |
139 146
|
imbi12d |
⊢ ( 𝑥 = ( 𝑓 “ ( 1 ... 𝑛 ) ) → ( ( 𝑥 ⊆ 𝐴 → ∀ 𝑘 ∈ ( 𝐴 ∖ 𝑥 ) ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ 𝑥 ) ) } ) ) ↔ ( ( 𝑓 “ ( 1 ... 𝑛 ) ) ⊆ 𝐴 → ∀ 𝑘 ∈ ( 𝐴 ∖ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) ) ) ) |
148 |
138 147
|
spcv |
⊢ ( ∀ 𝑥 ( 𝑥 ⊆ 𝐴 → ∀ 𝑘 ∈ ( 𝐴 ∖ 𝑥 ) ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ 𝑥 ) ) } ) ) → ( ( 𝑓 “ ( 1 ... 𝑛 ) ) ⊆ 𝐴 → ∀ 𝑘 ∈ ( 𝐴 ∖ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) ) ) |
149 |
132 136 148
|
sylc |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ∀ 𝑘 ∈ ( 𝐴 ∖ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) ) |
150 |
|
ffvelrn |
⊢ ( ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐴 ∧ ( 𝑛 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( 𝑓 ‘ ( 𝑛 + 1 ) ) ∈ 𝐴 ) |
151 |
77 119 150
|
syl2an |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑓 ‘ ( 𝑛 + 1 ) ) ∈ 𝐴 ) |
152 |
|
fzp1nel |
⊢ ¬ ( 𝑛 + 1 ) ∈ ( 1 ... 𝑛 ) |
153 |
75
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1→ 𝐴 ) |
154 |
119
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑛 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) |
155 |
|
f1elima |
⊢ ( ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1→ 𝐴 ∧ ( 𝑛 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ∧ ( 1 ... 𝑛 ) ⊆ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝑓 ‘ ( 𝑛 + 1 ) ) ∈ ( 𝑓 “ ( 1 ... 𝑛 ) ) ↔ ( 𝑛 + 1 ) ∈ ( 1 ... 𝑛 ) ) ) |
156 |
153 154 108 155
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝑓 ‘ ( 𝑛 + 1 ) ) ∈ ( 𝑓 “ ( 1 ... 𝑛 ) ) ↔ ( 𝑛 + 1 ) ∈ ( 1 ... 𝑛 ) ) ) |
157 |
152 156
|
mtbiri |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ¬ ( 𝑓 ‘ ( 𝑛 + 1 ) ) ∈ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) |
158 |
151 157
|
eldifd |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑓 ‘ ( 𝑛 + 1 ) ) ∈ ( 𝐴 ∖ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) |
159 |
127 149 158
|
rspcdva |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝐹 ‘ ( 𝑓 ‘ ( 𝑛 + 1 ) ) ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) ) |
160 |
125 159
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐹 ∘ 𝑓 ) ‘ ( 𝑛 + 1 ) ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) ) |
161 |
138
|
a1i |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑓 “ ( 1 ... 𝑛 ) ) ∈ V ) |
162 |
11
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → 𝐻 : 𝐴 ⟶ 𝐵 ) |
163 |
162 136
|
fssresd |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) : ( 𝑓 “ ( 1 ... 𝑛 ) ) ⟶ 𝐵 ) |
164 |
13
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ran 𝐻 ⊆ ( 𝑍 ‘ ran 𝐻 ) ) |
165 |
|
resss |
⊢ ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ⊆ 𝐻 |
166 |
165
|
rnssi |
⊢ ran ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ⊆ ran 𝐻 |
167 |
4
|
cntzidss |
⊢ ( ( ran 𝐻 ⊆ ( 𝑍 ‘ ran 𝐻 ) ∧ ran ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ⊆ ran 𝐻 ) → ran ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ⊆ ( 𝑍 ‘ ran ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) ) |
168 |
164 166 167
|
sylancl |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ran ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ⊆ ( 𝑍 ‘ ran ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) ) |
169 |
104 57
|
eleqtrrdi |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → 𝑛 ∈ ℕ ) |
170 |
|
f1ores |
⊢ ( ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1→ 𝐴 ∧ ( 1 ... 𝑛 ) ⊆ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( 𝑓 ↾ ( 1 ... 𝑛 ) ) : ( 1 ... 𝑛 ) –1-1-onto→ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) |
171 |
153 108 170
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑓 ↾ ( 1 ... 𝑛 ) ) : ( 1 ... 𝑛 ) –1-1-onto→ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) |
172 |
|
f1of1 |
⊢ ( ( 𝑓 ↾ ( 1 ... 𝑛 ) ) : ( 1 ... 𝑛 ) –1-1-onto→ ( 𝑓 “ ( 1 ... 𝑛 ) ) → ( 𝑓 ↾ ( 1 ... 𝑛 ) ) : ( 1 ... 𝑛 ) –1-1→ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) |
173 |
171 172
|
syl |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑓 ↾ ( 1 ... 𝑛 ) ) : ( 1 ... 𝑛 ) –1-1→ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) |
174 |
|
suppssdm |
⊢ ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) supp 0 ) ⊆ dom ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) |
175 |
|
dmres |
⊢ dom ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) = ( ( 𝑓 “ ( 1 ... 𝑛 ) ) ∩ dom 𝐻 ) |
176 |
175
|
a1i |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → dom ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) = ( ( 𝑓 “ ( 1 ... 𝑛 ) ) ∩ dom 𝐻 ) ) |
177 |
174 176
|
sseqtrid |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) supp 0 ) ⊆ ( ( 𝑓 “ ( 1 ... 𝑛 ) ) ∩ dom 𝐻 ) ) |
178 |
|
inss1 |
⊢ ( ( 𝑓 “ ( 1 ... 𝑛 ) ) ∩ dom 𝐻 ) ⊆ ( 𝑓 “ ( 1 ... 𝑛 ) ) |
179 |
|
df-ima |
⊢ ( 𝑓 “ ( 1 ... 𝑛 ) ) = ran ( 𝑓 ↾ ( 1 ... 𝑛 ) ) |
180 |
179
|
a1i |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑓 “ ( 1 ... 𝑛 ) ) = ran ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) |
181 |
178 180
|
sseqtrid |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝑓 “ ( 1 ... 𝑛 ) ) ∩ dom 𝐻 ) ⊆ ran ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) |
182 |
177 181
|
sstrd |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) supp 0 ) ⊆ ran ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) |
183 |
|
eqid |
⊢ ( ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) supp 0 ) = ( ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) supp 0 ) |
184 |
1 2 3 4 102 161 163 168 169 173 182 183
|
gsumval3 |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) = ( seq 1 ( + , ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) ) ‘ 𝑛 ) ) |
185 |
179
|
eqimss2i |
⊢ ran ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ⊆ ( 𝑓 “ ( 1 ... 𝑛 ) ) |
186 |
|
cores |
⊢ ( ran ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ⊆ ( 𝑓 “ ( 1 ... 𝑛 ) ) → ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) = ( 𝐻 ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) ) |
187 |
185 186
|
ax-mp |
⊢ ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) = ( 𝐻 ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) |
188 |
|
resco |
⊢ ( ( 𝐻 ∘ 𝑓 ) ↾ ( 1 ... 𝑛 ) ) = ( 𝐻 ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) |
189 |
187 188
|
eqtr4i |
⊢ ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) = ( ( 𝐻 ∘ 𝑓 ) ↾ ( 1 ... 𝑛 ) ) |
190 |
189
|
fveq1i |
⊢ ( ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) ‘ 𝑘 ) = ( ( ( 𝐻 ∘ 𝑓 ) ↾ ( 1 ... 𝑛 ) ) ‘ 𝑘 ) |
191 |
|
fvres |
⊢ ( 𝑘 ∈ ( 1 ... 𝑛 ) → ( ( ( 𝐻 ∘ 𝑓 ) ↾ ( 1 ... 𝑛 ) ) ‘ 𝑘 ) = ( ( 𝐻 ∘ 𝑓 ) ‘ 𝑘 ) ) |
192 |
190 191
|
eqtrid |
⊢ ( 𝑘 ∈ ( 1 ... 𝑛 ) → ( ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) ‘ 𝑘 ) = ( ( 𝐻 ∘ 𝑓 ) ‘ 𝑘 ) ) |
193 |
192
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑘 ∈ ( 1 ... 𝑛 ) ) → ( ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) ‘ 𝑘 ) = ( ( 𝐻 ∘ 𝑓 ) ‘ 𝑘 ) ) |
194 |
104 193
|
seqfveq |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( seq 1 ( + , ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) ) ‘ 𝑛 ) = ( seq 1 ( + , ( 𝐻 ∘ 𝑓 ) ) ‘ 𝑛 ) ) |
195 |
184 194
|
eqtr2d |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( seq 1 ( + , ( 𝐻 ∘ 𝑓 ) ) ‘ 𝑛 ) = ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) ) |
196 |
|
fvex |
⊢ ( seq 1 ( + , ( 𝐻 ∘ 𝑓 ) ) ‘ 𝑛 ) ∈ V |
197 |
196
|
elsn |
⊢ ( ( seq 1 ( + , ( 𝐻 ∘ 𝑓 ) ) ‘ 𝑛 ) ∈ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ↔ ( seq 1 ( + , ( 𝐻 ∘ 𝑓 ) ) ‘ 𝑛 ) = ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) ) |
198 |
195 197
|
sylibr |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( seq 1 ( + , ( 𝐻 ∘ 𝑓 ) ) ‘ 𝑛 ) ∈ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) |
199 |
3 4
|
cntzi |
⊢ ( ( ( ( 𝐹 ∘ 𝑓 ) ‘ ( 𝑛 + 1 ) ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) ∧ ( seq 1 ( + , ( 𝐻 ∘ 𝑓 ) ) ‘ 𝑛 ) ∈ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) → ( ( ( 𝐹 ∘ 𝑓 ) ‘ ( 𝑛 + 1 ) ) + ( seq 1 ( + , ( 𝐻 ∘ 𝑓 ) ) ‘ 𝑛 ) ) = ( ( seq 1 ( + , ( 𝐻 ∘ 𝑓 ) ) ‘ 𝑛 ) + ( ( 𝐹 ∘ 𝑓 ) ‘ ( 𝑛 + 1 ) ) ) ) |
200 |
160 198 199
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( ( 𝐹 ∘ 𝑓 ) ‘ ( 𝑛 + 1 ) ) + ( seq 1 ( + , ( 𝐻 ∘ 𝑓 ) ) ‘ 𝑛 ) ) = ( ( seq 1 ( + , ( 𝐻 ∘ 𝑓 ) ) ‘ 𝑛 ) + ( ( 𝐹 ∘ 𝑓 ) ‘ ( 𝑛 + 1 ) ) ) ) |
201 |
200
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( seq 1 ( + , ( 𝐻 ∘ 𝑓 ) ) ‘ 𝑛 ) + ( ( 𝐹 ∘ 𝑓 ) ‘ ( 𝑛 + 1 ) ) ) = ( ( ( 𝐹 ∘ 𝑓 ) ‘ ( 𝑛 + 1 ) ) + ( seq 1 ( + , ( 𝐻 ∘ 𝑓 ) ) ‘ 𝑛 ) ) ) |
202 |
1 3 102 115 118 121 123 201
|
mnd4g |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( ( seq 1 ( + , ( 𝐹 ∘ 𝑓 ) ) ‘ 𝑛 ) + ( seq 1 ( + , ( 𝐻 ∘ 𝑓 ) ) ‘ 𝑛 ) ) + ( ( ( 𝐹 ∘ 𝑓 ) ‘ ( 𝑛 + 1 ) ) + ( ( 𝐻 ∘ 𝑓 ) ‘ ( 𝑛 + 1 ) ) ) ) = ( ( ( seq 1 ( + , ( 𝐹 ∘ 𝑓 ) ) ‘ 𝑛 ) + ( ( 𝐹 ∘ 𝑓 ) ‘ ( 𝑛 + 1 ) ) ) + ( ( seq 1 ( + , ( 𝐻 ∘ 𝑓 ) ) ‘ 𝑛 ) + ( ( 𝐻 ∘ 𝑓 ) ‘ ( 𝑛 + 1 ) ) ) ) ) |
203 |
55 55 58 80 84 101 202
|
seqcaopr3 |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( seq 1 ( + , ( ( 𝐹 ∘f + 𝐻 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝑊 ) ) = ( ( seq 1 ( + , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝑊 ) ) + ( seq 1 ( + , ( 𝐻 ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝑊 ) ) ) ) |
204 |
54 59 81 87 87 89
|
off |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( 𝐹 ∘f + 𝐻 ) : 𝐴 ⟶ 𝐵 ) |
205 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ran ( 𝐹 ∘f + 𝐻 ) ⊆ ( 𝑍 ‘ ran ( 𝐹 ∘f + 𝐻 ) ) ) |
206 |
51 113
|
sylan |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ ( 𝑘 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ) → ( 𝑘 + 𝑥 ) ∈ 𝐵 ) |
207 |
206 59 81 87 87 89
|
off |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( 𝐹 ∘f + 𝐻 ) : 𝐴 ⟶ 𝐵 ) |
208 |
|
eldifi |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ ran 𝑓 ) → 𝑥 ∈ 𝐴 ) |
209 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) ) |
210 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝐻 ‘ 𝑥 ) = ( 𝐻 ‘ 𝑥 ) ) |
211 |
85 86 87 87 89 209 210
|
ofval |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐹 ∘f + 𝐻 ) ‘ 𝑥 ) = ( ( 𝐹 ‘ 𝑥 ) + ( 𝐻 ‘ 𝑥 ) ) ) |
212 |
208 211
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑥 ∈ ( 𝐴 ∖ ran 𝑓 ) ) → ( ( 𝐹 ∘f + 𝐻 ) ‘ 𝑥 ) = ( ( 𝐹 ‘ 𝑥 ) + ( 𝐻 ‘ 𝑥 ) ) ) |
213 |
24
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( 𝐹 supp 0 ) ⊆ ( ( 𝐹 ∪ 𝐻 ) supp 0 ) ) |
214 |
|
f1ofo |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 → 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –onto→ 𝑊 ) |
215 |
|
forn |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –onto→ 𝑊 → ran 𝑓 = 𝑊 ) |
216 |
214 215
|
syl |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 → ran 𝑓 = 𝑊 ) |
217 |
216 9
|
eqtrdi |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 → ran 𝑓 = ( ( 𝐹 ∪ 𝐻 ) supp 0 ) ) |
218 |
217
|
sseq2d |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 → ( ( 𝐹 supp 0 ) ⊆ ran 𝑓 ↔ ( 𝐹 supp 0 ) ⊆ ( ( 𝐹 ∪ 𝐻 ) supp 0 ) ) ) |
219 |
218
|
ad2antll |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( ( 𝐹 supp 0 ) ⊆ ran 𝑓 ↔ ( 𝐹 supp 0 ) ⊆ ( ( 𝐹 ∪ 𝐻 ) supp 0 ) ) ) |
220 |
213 219
|
mpbird |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( 𝐹 supp 0 ) ⊆ ran 𝑓 ) |
221 |
21
|
a1i |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → 0 ∈ V ) |
222 |
59 220 87 221
|
suppssr |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑥 ∈ ( 𝐴 ∖ ran 𝑓 ) ) → ( 𝐹 ‘ 𝑥 ) = 0 ) |
223 |
33
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( 𝐻 supp 0 ) ⊆ ( ( 𝐻 ∪ 𝐹 ) supp 0 ) ) |
224 |
223 35
|
sseqtrrdi |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( 𝐻 supp 0 ) ⊆ ( ( 𝐹 ∪ 𝐻 ) supp 0 ) ) |
225 |
217
|
sseq2d |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 → ( ( 𝐻 supp 0 ) ⊆ ran 𝑓 ↔ ( 𝐻 supp 0 ) ⊆ ( ( 𝐹 ∪ 𝐻 ) supp 0 ) ) ) |
226 |
225
|
ad2antll |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( ( 𝐻 supp 0 ) ⊆ ran 𝑓 ↔ ( 𝐻 supp 0 ) ⊆ ( ( 𝐹 ∪ 𝐻 ) supp 0 ) ) ) |
227 |
224 226
|
mpbird |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( 𝐻 supp 0 ) ⊆ ran 𝑓 ) |
228 |
81 227 87 221
|
suppssr |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑥 ∈ ( 𝐴 ∖ ran 𝑓 ) ) → ( 𝐻 ‘ 𝑥 ) = 0 ) |
229 |
222 228
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑥 ∈ ( 𝐴 ∖ ran 𝑓 ) ) → ( ( 𝐹 ‘ 𝑥 ) + ( 𝐻 ‘ 𝑥 ) ) = ( 0 + 0 ) ) |
230 |
19
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑥 ∈ ( 𝐴 ∖ ran 𝑓 ) ) → ( 0 + 0 ) = 0 ) |
231 |
212 229 230
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ∧ 𝑥 ∈ ( 𝐴 ∖ ran 𝑓 ) ) → ( ( 𝐹 ∘f + 𝐻 ) ‘ 𝑥 ) = 0 ) |
232 |
207 231
|
suppss |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( ( 𝐹 ∘f + 𝐻 ) supp 0 ) ⊆ ran 𝑓 ) |
233 |
|
ovex |
⊢ ( 𝐹 ∘f + 𝐻 ) ∈ V |
234 |
233 137
|
coex |
⊢ ( ( 𝐹 ∘f + 𝐻 ) ∘ 𝑓 ) ∈ V |
235 |
|
suppimacnv |
⊢ ( ( ( ( 𝐹 ∘f + 𝐻 ) ∘ 𝑓 ) ∈ V ∧ 0 ∈ V ) → ( ( ( 𝐹 ∘f + 𝐻 ) ∘ 𝑓 ) supp 0 ) = ( ◡ ( ( 𝐹 ∘f + 𝐻 ) ∘ 𝑓 ) “ ( V ∖ { 0 } ) ) ) |
236 |
235
|
eqcomd |
⊢ ( ( ( ( 𝐹 ∘f + 𝐻 ) ∘ 𝑓 ) ∈ V ∧ 0 ∈ V ) → ( ◡ ( ( 𝐹 ∘f + 𝐻 ) ∘ 𝑓 ) “ ( V ∖ { 0 } ) ) = ( ( ( 𝐹 ∘f + 𝐻 ) ∘ 𝑓 ) supp 0 ) ) |
237 |
234 21 236
|
mp2an |
⊢ ( ◡ ( ( 𝐹 ∘f + 𝐻 ) ∘ 𝑓 ) “ ( V ∖ { 0 } ) ) = ( ( ( 𝐹 ∘f + 𝐻 ) ∘ 𝑓 ) supp 0 ) |
238 |
1 2 3 4 51 87 204 205 56 75 232 237
|
gsumval3 |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( 𝐺 Σg ( 𝐹 ∘f + 𝐻 ) ) = ( seq 1 ( + , ( ( 𝐹 ∘f + 𝐻 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝑊 ) ) ) |
239 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ran 𝐹 ⊆ ( 𝑍 ‘ ran 𝐹 ) ) |
240 |
|
eqid |
⊢ ( ( 𝐹 ∘ 𝑓 ) supp 0 ) = ( ( 𝐹 ∘ 𝑓 ) supp 0 ) |
241 |
1 2 3 4 51 87 59 239 56 75 220 240
|
gsumval3 |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( 𝐺 Σg 𝐹 ) = ( seq 1 ( + , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝑊 ) ) ) |
242 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ran 𝐻 ⊆ ( 𝑍 ‘ ran 𝐻 ) ) |
243 |
|
eqid |
⊢ ( ( 𝐻 ∘ 𝑓 ) supp 0 ) = ( ( 𝐻 ∘ 𝑓 ) supp 0 ) |
244 |
1 2 3 4 51 87 81 242 56 75 227 243
|
gsumval3 |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( 𝐺 Σg 𝐻 ) = ( seq 1 ( + , ( 𝐻 ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝑊 ) ) ) |
245 |
241 244
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) = ( ( seq 1 ( + , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝑊 ) ) + ( seq 1 ( + , ( 𝐻 ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝑊 ) ) ) ) |
246 |
203 238 245
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) → ( 𝐺 Σg ( 𝐹 ∘f + 𝐻 ) ) = ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) ) |
247 |
246
|
expr |
⊢ ( ( 𝜑 ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ ) → ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 → ( 𝐺 Σg ( 𝐹 ∘f + 𝐻 ) ) = ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) ) ) |
248 |
247
|
exlimdv |
⊢ ( ( 𝜑 ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ ) → ( ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 → ( 𝐺 Σg ( 𝐹 ∘f + 𝐻 ) ) = ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) ) ) |
249 |
248
|
expimpd |
⊢ ( 𝜑 → ( ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) → ( 𝐺 Σg ( 𝐹 ∘f + 𝐻 ) ) = ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) ) ) |
250 |
7 8
|
fsuppun |
⊢ ( 𝜑 → ( ( 𝐹 ∪ 𝐻 ) supp 0 ) ∈ Fin ) |
251 |
9 250
|
eqeltrid |
⊢ ( 𝜑 → 𝑊 ∈ Fin ) |
252 |
|
fz1f1o |
⊢ ( 𝑊 ∈ Fin → ( 𝑊 = ∅ ∨ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ) |
253 |
251 252
|
syl |
⊢ ( 𝜑 → ( 𝑊 = ∅ ∨ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto→ 𝑊 ) ) ) |
254 |
50 249 253
|
mpjaod |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝐹 ∘f + 𝐻 ) ) = ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) ) |